Abstract
We explore the Mellin representation of correlation functions in conformal field theories in the weak coupling regime. We provide a complete proof for a set of Feynman rules to write the Mellin amplitude for a general tree level Feynman diagram involving only scalar operators. We find a factorised form involving beta functions associated to the propagators, similar to tree level Feynman rules in momentum space for ordinary QFTs. We also briefly consider the case where a generic scalar perturbation of the free CFT breaks conformal invariance. Mellin space still has some utility and one can consider non-conformal Mellin representations. In this context, we find that the beta function corresponding to conformal propagator uplifts to a hypergeometric function.
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References
G. Mack, D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE].
G. Mack, D-dimensional Conformal Field Theories with anomalous dimensions as Dual Resonance Models, Bulg. J. Phys. 36 (2009) 214 [arXiv:0909.1024] [INSPIRE].
K. Symanzik, On Calculations in conformal invariant field theories, Lett. Nuovo Cim. 3 (1972) 734 [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
M.F. Paulos, Towards Feynman rules for Mellin amplitudes, JHEP 10 (2011) 074 [arXiv:1107.1504] [INSPIRE].
D. Nandan, A. Volovich and C. Wen, On Feynman Rules for Mellin Amplitudes in AdS/CFT, JHEP 05 (2012) 129 [arXiv:1112.0305] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A Natural Language for AdS/CFT Correlators, JHEP 11 (2011) 095 [arXiv:1107.1499] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, AdS Field Theory from Conformal Field Theory, JHEP 02 (2013) 054 [arXiv:1208.0337] [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks, JHEP 01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, Analyticity and the Holographic S-matrix, JHEP 10 (2012) 127 [arXiv:1111.6972] [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, Unitarity and the Holographic S-matrix, JHEP 10 (2012) 032 [arXiv:1112.4845] [INSPIRE].
V. Gonçalves, J. Penedones and E. Trevisani, Factorization of Mellin amplitudes, JHEP 10 (2015) 040 [arXiv:1410.4185] [INSPIRE].
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix Bootstrap I: QFT in AdS, arXiv:1607.06109 [INSPIRE].
D.A. Lowe, Mellin transforming the minimal model CFTs: AdS/CFT at strong curvature, Phys. Lett. B 760 (2016) 494 [arXiv:1602.05613] [INSPIRE].
S. Stieberger and T.R. Taylor, Superstring Amplitudes as a Mellin Transform of Supergravity, Nucl. Phys. B 873 (2013) 65 [arXiv:1303.1532] [INSPIRE].
M.F. Paulos, M. Spradlin and A. Volovich, Mellin Amplitudes for Dual Conformal Integrals, JHEP 08 (2012) 072 [arXiv:1203.6362] [INSPIRE].
D. Nandan, M.F. Paulos, M. Spradlin and A. Volovich, Star Integrals, Convolutions and Simplices, JHEP 05 (2013) 105 [arXiv:1301.2500] [INSPIRE].
V. Gonçalves, Four point function of \( \mathcal{N} \) = 4 stress-tensor multiplet at strong coupling, JHEP 04 (2015) 150 [arXiv:1411.1675] [INSPIRE].
L.F. Alday, A. Bissi and T. Lukowski, Lessons from crossing symmetry at large-N , JHEP 06 (2015) 074 [arXiv:1410.4717] [INSPIRE].
A.V. Belitsky, S. Hohenegger, G.P. Korchemsky, E. Sokatchev and A. Zhiboedov, From correlation functions to event shapes, Nucl. Phys. B 884 (2014) 305 [arXiv:1309.0769] [INSPIRE].
A.V. Belitsky, S. Hohenegger, G.P. Korchemsky, E. Sokatchev and A. Zhiboedov, Event shapes in \( \mathcal{N} \) = 4 super-Yang-Mills theory, Nucl. Phys. B 884 (2014) 206 [arXiv:1309.1424] [INSPIRE].
A.V. Belitsky, S. Hohenegger, G.P. Korchemsky, E. Sokatchev and A. Zhiboedov, Energy-Energy Correlations in N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 112 (2014) 071601 [arXiv:1311.6800] [INSPIRE].
L.F. Alday and A. Bissi, Unitarity and positivity constraints for CFT at large central charge, arXiv:1606.09593 [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
Y. Nakayama, Scale invariance vs conformal invariance, Phys. Rept. 569 (2015) 1 [arXiv:1302.0884] [INSPIRE].
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ArXiv ePrint: 1607.07334
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Nizami, A.A., Rudra, A., Sarkar, S. et al. Exploring perturbative conformal field theory in Mellin space. J. High Energ. Phys. 2017, 102 (2017). https://doi.org/10.1007/JHEP01(2017)102
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DOI: https://doi.org/10.1007/JHEP01(2017)102